Many locales with potential have restricted access by the authorities, but this method may have a larger use.

Imagine an alternative ecological movement, a research center, or a scientific foundation (or all three). Especially those with strict ecological preferences may find it easy. A worthy mission, and especially important results coming from the research, can bring recognition by themselves. And an 'excuse' of sorts for the presence of people where they are not expected, or doing things not intended, is very useful.

It would be off the beaten path somewhere, in any location suitable for micronational efforts (abandoned or disputed land, unsettled island, etc.). The group behind it may choose any fitting topic of research and/or any additional activities, like:

the usual marine biology thing, ideal for islands, which allows you to stare into the sea all day, do some fishing, and doesn't hinder using the land. Substitute for any other biology if necessary.

cleaning up the 'viroment - there's islands where sea currents deposit lots of trash, wreckage, nets, and so on. There are also dumps of dangerous materials, abandoned bases and the like, if that is your taste. What? It will take decades to clean all that mess? Oh well. Then we'll have to stay around for decades. (One article about the topic.)

restoring corral from damage or pollution, which can take a LOT of time… plus it tends to increase the size of an island over time, which can't be bad.

alternative/ecological/green/whatever housing or lifestyle with a minimal impact on the environment. Here you can use all the weird designs and plastic bottles… the stranger they are, the better excuse they provide.

saving an endangered species, and protecting it all the time

Now if you are to stay there for years, it is understandable you will want to have your family around. With various support personnel, few people might actually end up doing the research itself.

And it is of course very expensive to keep importing all the necessities. Unless you mess up the surroundings too much, no one can blame you for trying to be self-sufficient.

On the other hand, it could be possible to get funding from other organizations - but be aware that the more they invest, the more control and/or results they expect in return. If the basic organization is a foundation, and it has a sufficiently good image, donations may be also a good source of income. Chalk down here any exchange students, scientists, and other forms of cooperation and support.

Requirements:

needs an institutional backbone, registered foundations/non-profits in one or more countries

carefully research the relevant laws and rules to make sure there is minimal interference from the authorities

needs to keep up the researching image while creating a viable settlement. The public image and contacts with scientific institutions can be very useful.

A research station is likely to be treated very differently than a separatist movement. It could have a legitimate presence even on disputed lands, be in stable contact with and hopefully have a good image by The Powers That Be, and with some luck be exempt from taxes given the nonprofit nature of the enterprise. If the place is outlying enough, you will make your own laws.

Staying for a few years, and building up a veritable colony could be considered a claim. Many micronations promise themselves to build up a modern economy, sporting advanced research and stuff… maybe it could be possible to start with it.